Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion

Numerical approximation of 2D Fredholm integral eigenvalue problems by orthogonal wavelets

“…We refer to [2,21,22] for details on the computation of exact eigenvalues for these kernels. Kernel (23) is a common benchmark for Fredholm integral eigenvalue problems [1,4,5,[22][23][24]. S.P.…”

“…The recent popularity of numerical methods for partial differential equations with random input data based on the Karhunen-Loève expansion of random fields has vitalized the research on numerical methods for Fredholm integral eigenvalue equations [1][2][3]. One of these techniques is the Galerkin method with wavelet basis functions, which has been thoroughly studied in the particular case of the Haar basis [4,5].…”

Analysis of Uncertainty and Variability in Finite Element Computational Models for Biomedical Engineering: Characterization and Propagation

“…Given the domain D and the sample space Ω, this approach, based on model reduction techniques, expands any random field, $k\left(xMathClass-punc;\omega \right)MathClass-punc:DMathClass-bin\times \mathrm{\Omega }MathClass-rel\to \mathrm{double-struckR}$, in a sum of products of functions of the stochastic and deterministic parameters in a Fourier-type series. In this way, a random variable is defined within the random field k at a certain point x ∈ D , thus a real number k ( x , ω) is obtained for each realization ω ∈ Ω (Betz et al, 2004; Eiermann et al, 2007; Sachdeva et al, 2007). …”

Large scale random fields generation using localized Karhunen–Loève expansion

“…Other methods, described and compared in [26] can be used to approximate the eigenfunctions more efficiently, such as collocation and Galerkin integration [26]. In this case, the eigen-functions are approximated by a set ofñ s < n s basis functions, leading tõ n s ×ñ s matrix generalized eigen-problem whose complexity is O(2ñ 3 s ).…”